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          ML Summary 7-Anomaly Detection and Recommendation Systems
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        <h1 id="异常检测"><a href="#异常检测" class="headerlink" title="异常检测"></a>异常检测</h1><p>通过现有正常样本<strong>拟合</strong>出高斯分布，当未知样本出现时，算出它出现的概率，若概率值小于设定值，则判定为异常样本。当数据不是标准高斯分布时，可先将数据转换为标准高斯分布（归一化及放缩），再将其送入异常检测算法。对于样本数据的均值和方差，计算公式为：</p>
<p>均值的公式：<br>$$<br>\mu_i&#x3D;{1\over m}\Sigma_{j&#x3D;1}^mx_i^{(j)}<br>$$<br>方差的公式：<br>$$<br>\sigma_i^2&#x3D;{1\over m}\Sigma_{i&#x3D;1}^m(x_i^{(j)}-\mu_i)^2<br>$$<br>当样本数量大于特征数量时，可以选用多元高斯分布，多元高斯分布相比于原分布增加了协方差矩阵Sigma，Sigma对高斯分布的影响如下图所示。</p>
<p><img src="https://s2.loli.net/2022/01/22/mwbaXYrONVDBGoS.png" alt="多元高斯.png"></p>
<p>当样本数量大于特征数量时，可以选用多元高斯分布。原高斯分布如果想要体现特征之间的特征，需要额外组合出新的特征，多元高斯分布则可以直接找出特征间的关系。原高斯分布与多元高斯分布对比图如下。</p>
<p><img src="https://s2.loli.net/2022/01/22/NYDr58wLtPm3QMb.png" alt="多元高斯与原高斯对比.png"></p>
<p>在异常检测数据集分配中，正常样本和异常样本都需分配到交叉验证数据集和测试数据集中，这两个数据集最好不要用相同的数据。</p>
<h1 id="推荐系统"><a href="#推荐系统" class="headerlink" title="推荐系统"></a>推荐系统</h1><h2 id="协同过滤算法"><a href="#协同过滤算法" class="headerlink" title="协同过滤算法"></a>协同过滤算法</h2><p>可以用theta和x交替优化，求出最优的theta和特征值x。如先用theta求出最优的特征x，再用特征x拟合出最佳theta，如此循环计算，最后算法收敛于最优的theta和x。</p>
<p>为方便计算，将theta和x组合到同一个代价函数中，以便同时最小化theta和特征x。<strong>没有正则化</strong>时的代价函数为<br>$$<br>J(x^{(1)},…,x^{(n_m)},\theta^{(1)},…,\theta^{(n_u)})&#x3D;{1\over 2}\Sigma_{(i,j):r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})^2<br>$$<br>代价函数的对x和theta的偏导数为<br>$$<br>{\part J \over \part x_k^{(i)}}&#x3D;\Sigma_{j:r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})\theta_k^{(j)}<br>$$</p>
<p>$$<br>{\part J \over \part \theta_k^{(i)}}&#x3D;\Sigma_{j:r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})x_k^{(j)}<br>$$</p>
<p><strong>正则化</strong>后的代价函数为<br>$$<br>J(x^{(1)},…,x^{(n_m)},\theta^{(1)},…,\theta^{(n_u)})&#x3D;{1\over 2}\Sigma_{(i,j):r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})^2+({\lambda\over 2}\Sigma_{j&#x3D;1}^{n_u}\Sigma_{k&#x3D;1}^n(\theta_k^{(j)})^2)+({\lambda\over 2}\Sigma_{i&#x3D;1}^{n_m}\Sigma_{k&#x3D;1}^n(x_k^{(j)})^2)<br>$$<br>代价函数的对x和theta的偏导数为<br>$$<br>{\part J \over \part x_k^{(i)}}&#x3D;\Sigma_{j:r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})x_k^{(j)}+\lambda x_k^{(j)}<br>$$</p>
<p>$$<br>{\part J \over \part \theta_k^{(i)}}&#x3D;\Sigma_{j:r(i,j)&#x3D;1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})x_k^{(j)}+\lambda\theta_k^{(j)}<br>$$</p>
<h1 id="Tips"><a href="#Tips" class="headerlink" title="Tips"></a>Tips</h1><ul>
<li>在电影推荐系统中，通常默认为用户在0到5分之间打分。如果其他的推荐系统不在固定范围内评分，需要将评分进行放缩及归一化处理。示例如下<br><img src="https://s2.loli.net/2022/01/22/4T9vUqXufMzVjx3.png" alt="评分系统归一化.png"></li>
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